Question

in $\triangle mno$, $\overline{mo}$ is extended through point $o$ to point $p$, $m\angle omn = (3x + 14)\degree$, $m\angle mno = (2x - 3)\degree$, and $m\angle nop = (8x - 19)\degree$. what is the value of $x$?
answer attempt 1 out of 2
$x = $

Explanation:

Step1: Recall Exterior Angle Theorem

The exterior angle theorem states that an exterior angle of a triangle is equal to the sum of the two non - adjacent interior angles. In $\triangle MNO$, $\angle NOP$ is an exterior angle, and the two non - adjacent interior angles are $\angle OMN$ and $\angle MNO$. So, $m\angle NOP=m\angle OMN + m\angle MNO$.

Step2: Substitute the given angle measures

We know that $m\angle OMN=(3x + 14)^{\circ}$, $m\angle MNO=(2x-3)^{\circ}$, and $m\angle NOP=(8x - 19)^{\circ}$. Substituting these into the equation from the exterior angle theorem, we get:
$$8x-19=(3x + 14)+(2x-3)$$

Step3: Simplify the right - hand side of the equation

First, combine like terms on the right - hand side. $(3x+14)+(2x - 3)=3x+2x+14 - 3=5x + 11$. So our equation becomes:
$$8x-19=5x + 11$$

Step4: Solve for x

Subtract $5x$ from both sides of the equation:
$$8x-5x-19=5x-5x + 11$$
$$3x-19=11$$
Then, add 19 to both sides:
$$3x-19 + 19=11 + 19$$
$$3x=30$$
Divide both sides by 3:
$$x=\frac{30}{3}=10$$

Answer:

10